TABLE V.

PRECISE CALCULATION OF FUNCTIONS.

THE proportional parts, as given in Table IV, are sufficient for ordinary use. When precision is desired the following rules should be observed:

I. In finding the logarithmic function of an angle expressed in degrees, minutes, and seconds, derive it from that function which is nearest to it, whether greater or less; for, the proportional parts, being only approximations, should be multiplied by as small a number as possible.

II. In finding the angle from its given function, use that logarithm which differs least from the one given, subtracting or adding as the case may be.

III. To find the logarithmic sine of an angle of less than 2° 36′: reduce it to seconds; add the logarithm of the number of seconds to the logarithmic sine of one second, which is 4.685575; from this sum subtract the difference in the following table corresponding to the number of seconds; the remainder is the required logarithmic sine within one millionth.

IV. Conversely, to find the angle when the given logarithmic sine is less than 8.656702: first, find the angle approximately by Table IV; reduce this to seconds; add to the given sine the difference in the following table corresponding to the number of seconds; from this sum subtract 4.685575; the remainder is the logarithm of the required number of seconds within one.

V. To find the logarithmic tangent of an angle less than 2° 36′: reduce it to seconds; add to the logarithm of the number of seconds the logarithmic tangent of one second, which is 4.685575; to this sum add the difference in the table (p. 419 and 420) corresponding to the number of seconds; the sum is the required logarithmic tangent within one millionth.

VI. To find the angle when the given logarithmic tangent is less than 8.657149, which is the tangent of 2° 36′: first find the angle approximately by Table IV; reduce it to seconds; subtract from the given tangent the difference in the table corresponding to the number of seconds; from this remainder subtract 4.685575; the remainder is the logarithm of the required number of seconds within one.

VII. To find the logarithmic cotangent of an angle less than 2° 36: reduce it to seconds; subtract the logarithm of the number of seconds from the logarithmic cotangent of one second, which is 15.314425; from this remainder subtract the difference in the table corresponding to the number of seconds; the remainder is the required logarithmic cotangent within one millionth.

VIII. To find the angle when the given logarithmic cotangent is greater than 11.342851, the cotangent of 2° 36': first find the angle approximately by Table IV; reduce it to seconds; add to the given cotangent the difference in the table corresponding to the number of seconds; subtract this sum from 15.314425; the remainder is the logarithm of the required number of seconds within one.